A space-time variational formulation for the boundary integral equation in a 2D elastic crack problem

M2AN - M

E. B ÉCACHE

T. H A D UONG

A space-time variational formulation for the boundary integral equation in a 2D elastic crack problem

M2AN - Modélisation mathématique et analyse numérique, tome 28, n^o2 (1994), p. 141-176

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Article numérisé dans le cadre du programme

\ i \l\ \ \ MODÉUSATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE (Vol. 28, n° 2, 1994, p. 141 à 176)

A SPACE-TIME VARIATIONAL FORMULATION FOR THE BOUNDARY INTEGRAL EQUATION

IN A 2D ELASTIC CRACK PROBLEM (*)

by E. BÉCACHE C' 2) and T. H A DUONG (*• 3)

Communicated by P. G. CIARLET

Abstract. — This paper investigates the transient elastic wave scattering by a crack in R² by means of Boundary Intégral Equation Method. The analysis of the Laplace-Fourier transform (in time) of the intégral operator allows to obtain existence, uniqueness and continuity dependence of the solution with respect to the data, in a Sobolev functional framework. A régularisation of the hypersingular BIE is applied in order to remove the hypersingularity and to write the associated time-space variational formulation on a tractable form. A Galerkin-type approximation is then performed to solve this variational formulation and

we finally present some numerical results.

Résumé. — Nous nous intéressons à la résolution par une méthode d'équations intégrales d'un problème de diffraction d'ondes élastiques transitoires par une fissure. L'analyse de la transformée de F ourier-Laplace (en temps) de Vopérateur intégral permet d'obtenir des résultats d'existence, d'unicité et de continuité de la solution par rapport aux données dans des espaces fonctionnels de type Sobolev. Le noyau de V équation intégrale étant hypersingulier, on ne dispose pas directement d'une expression calculable de la forme bilinéaire associée. Cette difficulté est surmontée en appliquant une méthode de régularisation qui fournit finalement la formulation variationnelle espace-temps du problème. Cette formulation est enfin approchée à

l'aide d'un schéma de Galerkin et nous présentons quelques résultats numériques.

1. INTRODUCTION

This paper deals with the transient elastic scattering by a crack in R2. An important field of application of such problem is non-destructive évaluation research (for elastic materials). It consists in deducing the

(*) Manuscript received May 11, 1993.

C1) CMAP, Ecole Polytechnique, F-91128 Palaiseau Cedex.

(2) INRIA, Domaine de Voluceau-Rocquencourt, BP 105, F-78153 Le Chesnay Cedex.

(3) Université de Compiègne, F-60206 Compiègne Cedex.

M2 AN Modélisation mathématique et Analyse numérique 0764-583X/94/02/S 4.00

présence and characteristics of the crack in a material from the analysis of the scattered wave (inverse problem) and therefore it necessits a good knowledge of the direct problem. The BIEM (Boundary Intégral Equation Method) has been shown to give accurate numerical results in external problem s and is commonly applied in scattering problems with collocation type discretization (see e.g. Beskos [7], for a survey on these methods in elastodynamics) as well as with Galerkin type discretization (see e.g. Cortey-Dumont [10]).

Ho wever, until recently, the literature concerning the time-dependent B1E was rather poor. A mathematical analysis of time-dependent intégral équations has been developed these last years by Bamberger & Ha Duong [IL [2], [12], in the case of acoustic waves. We present here a généralisation of these results to the elastodynamic case. The time-dependent intégral operator is studied as a spatial pseudo-differential operator, with the frequency variable (ie. the Fourier-Laplace transform of the time variable) as a complex parameter. In the Sobolev type functional framework (linked to the elastic energy), we obtain the existence and uniqueness of the solution of the BIE and its continuous dependence with respect to the data.

A difficulty appears when we want to solve this BIE numerically, related to the double layer représentation of the solution. Indeed, the BIE has a hypersingular kernel. However, a lot of régularisation technics have been investigated these last years to evaluate the hypersingular intégral in the time domain (Nishimura et al. [21], Sladek & Sladek [24].,.) as well as in the frequency domain (cf. Martin & Rizzo [18], Krishnasamy et al, [15], Bui [9], Bonnet [8]...). Another method has been proposed by Nédélec [20], for the frequency problem based on the variational formulation and leads to weakly singular kernels (after intégrations by parts). We use in the present paper the régularisation technique proposed by Nishimura & Kobayashi [22], and extended by Bécache et al. [6], to anisotropic media, which is very gênerai and can be used with collocation as well as Galerkin type discretization.

This paper is organised as follows. We set in the next section the transient scattering problem and the associated Boundary Intégral Equation. The mathematical analysis of the intégral operator is done in section 3, through the analysis of the Fourier-Laplace transformed problem. The Fourier- Laplace transformed operator is shown to be an isomorphism in some Sobolev spaces. The knowledge of the dependence of the constants with respect to the frequency co allows to dérive the properties of the time- dependent intégral operator (and the time-dependent variational formulation) from the previous analysis. The main resuit of this section concerns the existence, the uniqueness and the continuity of the solution with respect to the data. The variational formulation has then to be explicited on a tractable form, in order to be implemented. We thus have to remove the hypersingula- rity of the BIE by using a régularisation method. This is the object of the

A 2D ELASTIC CRACK PROBLEM 143 section 4, where we describe the way to dérive the space-time variational formulation from the régularisation method developed in [6]. Finally, this problem is solved by means of a Galerkin type approximation in section 5 for a rectilinear crack. A comparison between numerical results obtained with different formulations ([5] and [21]) is then presented. We note that the formulation proposed hère yields more stable results than the one used in [5].

2. NOTATIONS AND GOVERNING EQUATIONS

Let F be an open curve representing a crack in a two-dimensional isotropic elastic medium and O = R2\F the scène where elastic waves are propaga- ting. We suppose that F is part of a simple, closed Jordan curve F, so that a normal vector n can be defined on F separating locally (except at the two end-points of F) the medium 12 into two disjoint sets, one of which is exterior to f and pointed in by n. We dénote by O_ {O + ) the interior (exterior) domain delimited by F, and Fo = F\F (see 2.1).

n

• = & * " * •

• • • • • . , • • '•>'•,'•

n.

Figure 2.1.

We consider the scattered wave problem :

p IJL _ div a (u) = 0 in / 3 x i ? (i) 2(x, O = — (x, O = 0 in H x R_ (ii)

<r (M) n = g on F x R (iii)

(2.1)

where ü is the displacement field, cr(u) the stress tensor which, in the isotropic case, is related to the déformation tensor e with the Hooke's law (2.2)

144 E. BÉCACHE, T. HA DUONG

and A? fx are the Lamé's constants, The given data g is due to an incident wave and we assume that it is also null for t *s 0. From the Betti's reciprocal theorem, it is known that a solution of (2.1) can be represented as a double layer elastic potential

j

u{x, t) = - J T(n(y))(pc, y, . j* [u(y, . )]dyy

where

(i) [ ƒ ] = ƒ _ — ƒ + is the jump of a function over f (ii) * dénotes the convolution in the time variable

(iii) T(n^y) = Xn(y) i.e. T^ij{n^y) = £^ik.j n^k(y) where £^ik;j - <r^ik(Üj) is the stress tensor of the fundamental displacement tensor Uj9 which is defined by (2.4) The analytic expression of the fundamental displacement tensor U = Uu = (Ö,.),. is given by

1

To obtain (2,3), it may be useful to consider u as the solution of interior and exterior elastic équation

in x R) U {iî_ x R) (2.5)

A 2D ELASTIC CRACK PROBLEM 145 with the boundary conditions

[S] = 0 on F^o xR (2.6)

[a(u)n] = 0 on F xR . (2.7) It is easy to show that from (2.6) and (2.7), the équation (2.5) is also satisfied for (x, t) G F^o x /?, thus the first équation of (2.1) is obtained again. Now, formula (2.3) yields the solution of (2.1) as soon as the jump function

<p — [ü] is known on F x R. From the boundary condition (2.1, iii),

<p satisfies the following boundary intégral équation on F x R

lim a i - ! T(n(y))(x,y,.)*$(y,.)dy^y) = g(x, t) (2.8)

x' =x+ en(x) \ J r f

Since formai permutation of dérivations and intégration in (2.8) gives rise to singular and hypersingular intégrais, we shall corne back to (2.5), (2.6), (2.7) to give another définition of the LHS of (2.8). Let us dénote it by D<p, then we have

where u satisfies (2.5), (2.7) and

_ ( 2 . . 0 ,

<p on F x R .

In the following sections, we study the operator D defined by (2.9), giving in particular a space-time variational formulation of problem (2.8) with only weakly singular intégrais. A weak form of well-posedness of équation (2.8) is also obtained securing the stability of calculations by the variational formulation. Numerical expérimentations will be shown confirming this.

3. MATHEMATICAL ANALYSIS

We first study the following problem

p ! ü _ d i v cr (M) = 0 in {fi , x R) U (12_ x R) (a) dt2

u(x9t) = — (x,t) = 0 for ; ^ 0 (b) dt

[(tr(u)n)] =0 on F xR (c) (cr(ü)n)⁺ = (<r(u)n)_ = g on F x R (d) (3.1)

Thus, we don't impose, at that time, the condition [ü] = 0 on Fo x R. Following the ideas of [1], [2] we start by Fourier-Laplace transforming the problem with respect to the time variable ans study the transformed problem in function of the dual variable. Results for (3.1) will be obtained by the inverse transform. To fix the notations, we dénote by

ƒ(*>)= f(t)el^dt (3.2)

JR

the Fourier transform for ƒ e Ll(R\ and extend it as usual to the complex half-plane 3m (cu ) >• 0 when ƒ is causal (i.e. f(t) = O for t <: 0). This Fourier-Laplace transform is also extended to les s regular functions or distributions ƒ, with values in a Hubert space (see [28]). This point of view is useful here, as we consider M as a function of t with values in functional spaces in x.

3.1. The Fourier-Laplace transform problem We consider the problem

div cr(u) + pü>2ü = 0 in O+ U / 2 „ (a)

[o-(u)n] = 0 on f (b) (3.3) (a(ü)n)+ = (cr(ü)n)_ = g on f . (c)

The équation (a) in (3.3) is clearly the transformation of the équation (a) in (3.1). The équation (b) of (3.1) will be taken into account when we look for the only solutions of (3.3) which satisfy the Paley-Wiener theorem for causal functions. Usually, the elastic équation in (3.3) is treated with a real frequency co and a radiation condition must be added to (3.3) to be well- posed. With a>f = 3m (o> ) :=*- 0, this radiation condition can be included in a functional framework, and we show below that (3.3) defines an isomorphism for ge (H~ m(r))2 to ü belonging to a closed subspace of (Hl(&+ U O__ )f, This can be done as in the work of [1], [2] for acoustic waves. However, to have a finer behavior of M, we will define ^-dépendent norms for these spaces. When F is a straight crack, these norms were introduced in [12]. For a curved crack, it is necessary to use a localisation trick. Consider a finite covering of f by open sets ((Pi )f e z, a smooth partition of unity (a, ) subordinate to this cover and diffeomorphisms (<p( ) transporting Oi to the open set Q = {- 1 <JC1S JC2 < 1} in R², with <Pii®i O T ) =

[\x^x | -c 1, x² = 0 } . Usual compatibility assumptions are done when d)ⁱ n &j ^ 0. Now, given a function ƒ defined on Z\ we set

(*,-ƒ) (*i) = («,• ƒ ) « <PT\xi> 0 ) - 1 < * ! < 1 . (3.4)

A 2D ELASTIC CRACK PROBLEM 147

Since supp at <= (Di and <pi is a diffeomorphism, it is clear that 6tf has its support included in ] - 1, + 1 [. Therefore, we identify this function with its extension by 0 out of the interval ]— 1, + 1 [ and can define

I/U,r =

It is clear that for every co ^ 0, | ƒ | ^ w ^ is a norm on Hs(F) equivalent to the usual norm. On the other hand, we define

II ƒ 1 1 , . . = ( M

+ \t\

)

\f(t)\

dï) (3.6)

for <o=£0, feH^s(R²) and ||/||Jj(ll>ffl the induce norm on H^s{&) if G is an open set in R2 :

II/ L, .. o =

{ II/ IL, „ ; ƒ e H°(R

), f

= ƒ} . (3.7)

For s = 1, one has

( £

)

|

„

(3.8)

where C l and C2 are positive constants independent of co, and we will use indifferently these two equivalent norms. Obviously, similar remarks can be made for positive integer s, but we need only the case 5 = 1 below. The first thing we have to do is to précise the dependence on <o of the usual trace theorems, when the norms (3.5) and (3.7) are used. Let (Pbea regular open set with compact boundary 90. We dénote by//J,((P) and H^2(b(9) the spaces Hl((9) and Hm(d(9) equipped with the norms (3.7) and (3.5) respectively.

LEMMA 3.1 : a) There exists a positive constant C depending only on 6 and tof such that

m

for all u ofH1 {(9) and co of the half-plane {3m (co) = co} 3= tof =>0}, where y dénotes the trace application.

b) On the other sense, one can construct an extension operator A from Hm(h<9) to Hl(<9) such that

/ 1 + «/ \ m

for all <p eHm(d(9) and co e {Jm (00) = coï ^ co?>0}.

vol. 28, n° 2, 1994

Proof ; The proof consists in similar calculations as for the usual norms (see [11]), so we only sketch out hère the steps.

(i) Using the atlas {(9^h <p⁽), one reduces the problem of the trace application in the half-space R\ = {(x^l7 x²) ; x² > 0 } .

(ii) In this case, a partial Fourier transform with respect to x1 can be carried out, leaving a one real variable problem to be dealed with. Foliowing exactly the path of Dautray-Lions's proof, one obtains the inequalities

with C independent od co, instead of (3.9) and (3.10).

(iii) However, in the process of step (i), one has to evaluate V{Ou) = u V0 + 0 Vu where 6 is a regular truncature function. Thus, in the expression \ a> {6u)\2 + \V{6u)\2, a non homogeneous term (with respect to

co) uV6 is introduced. The dependence on co of the continuity constants in (3.9) and (3.10) is now clear because of the inequality

C^l + C² \co\² 1 + cof

— ~—-**C •—- for \co\^col>§ (3.13)

| co \2 cof

where C ^x, C² are constants and C = max (C^x, C²). • An immédiat useful conséquence of this lemma concerns the analogous resuit with the jump operator, which appears in the following :

COROLLARY 3.1 : (of lemma 3.1J. Let (9_ (resp. ®+) a bounded regular domain (resp. its exterior domain), and let y~ and A~ {resp. y+ and A ⁺ ) the corresponding trace and extension operators. One defines the jump operator from Hl{(9_ U (9+)to Hm(d®_ ) as y = y' - y+ = [. ].

a) There exists a positive constant C depending only on (9 _ and O+ such that

for ail u ofH1 (0_ U 0+ ) and co ofthe half-plane {3m {co) = co l ^ co f > 0 }.

b) On the other sensé, one can construct an « extension » operator A from Hl/2{W_ ) to Hl{(9_ U <9+ ) such that yA<p = <p and

1/2

for ail <p e H^m{d(9_ ) and co e {3m {co) = coj ^ C U7° > 0 } .

A 2D ELASTIC CRACK PROBLEM 149

A second w-dependenee question concerns the Korn inequality. More precisely, we would like to compare the H^ (Ü ) norm of the elastic displacement ü with its energy

Ew( S ; / 2 ) ^ i p I I ^ H ^ + i \ <r{u).ê{ü)dx. (3.16) LEMMA 3.2 : For regular open sets f2 with compact boundary, we have

for all u e (H1^ (f2 ))25 where the constants C l and C2 depend only on f2 and on the elastic medium and (oI ^ cof>0.

Proof : The second inequality comes from the obvious || s(u)||2 =s || Vu ||2. The first one comes from the ellipticity assumption, which is always supposed for the elastic constants A, §JL : (o-(2), e(ü)) s* C ||^(w)||L and the Korn inequality

J | ; + | | 2 | | J ) (3.18)

Now, we come back to the problem (3.3).

THEOREM 3.1 : For every m in the half plane {fel (<o ) = ojj ^ o)f> 0 } , and g e (H"m(f)fi the problem (3.3) is well-posed in {Hl(f2+ U n_ ))2.

Proof : From the Green formula

f - f f

div o-(Ü).vdx = - a(u).e(v)dx+ <r(2) n . Ü dy (3.19)

JD JD JW

where n dénotes the exterior unit normal to D» applied to f2 + and Z2_ , it is easily seen that (3.3) admits the following variational formulation

j

ue (H^l{f2⁺ U f2_)f

(<r(Ü). s(v)~~ pa)2ü. v)dx = ^.[?]<iy (3.20)

f

u û^ J f

Vv e (Hï(f2+ U /2„))2.

Now, denoting the LHS of the equality in (3.20) by a(u9 v) — which is a

c o n t i n u o u s sesquilinear form on {Hl(fl+ U £1 _ ))2 — w e have

( o - ( w ) . s{u) + p \<o |2 | M2 + u /2_

= 2 * > , £ „ ( « , / 2+ U / 2 _ ) . (3.21) From (3.17), one gets the following coerciveness inequality satisfied by

\a(M,u)\»C —⁷ r—f - . (3.22)

1 1 + cof \<o\

The conclusion follows as usual, and one gets, using the trace estimate (3.9)

(1 + tof)312

H i i „

u « -

— z t ~

*-"'

-

for the solution ü of (3.3). • Remark 3.1 : It is known that from the Green formula (3.19) and the density of regular functions in the space Jf = {u e (H1 (O))2; div o-(u) e L2(/2)}, one can define for every ü e Jf its normal stress (T(Ü)n in (// ^{1 / 2}(6^))². It follows that the space

%„ = {Üe (H^l(f2⁺ U n_))²; p*>²w + div<r(5) = 0 ;

and [a{u)n] = 0}

is well defined as a closed subspace of (H^l(f2⁺ U 12 _ ))² and for u e 9C ^ its normal stress is defined as :

(<r(ü)n9 <?)_1 / 2 j l / 2 ; f = a(Ü,A$) V^ e (Hm(f)f (3.24) where A is the operator defined in corollary 3.1. Thus, by the Cauchy- Schwartz inequality and the estimâtes (3.15) and (3.17), one has

. (3.25)

We notice hère that theorem3.1 establishes an isomorphism between g e (ƒƒ" 1/2(JT))2 and ü e 9C^ unique solution of problem 3.3. However, in the estimate (3.23), which is the reciprocal estimate of the inequality (3.25)

A 2D ELASTIC CRACK PROBLEM 151 through this isomorphism, a factor | <o | is introduced. We will see below that this factor corresponds to a « lost of regularity » in the time variable.

Our next task is to study the Boundary Intégral Equation which allows to solve the problem (3.3). It is well known (see [19]), that, for regular function

$ on f, the double layer potential

« » ( * ) = - T<o&y){x-y)?(y)dy(y) xen+uf2_ ( 3 . 2 6 ) Jr

(where T^Çriy) is the normal stress tensor of the fundamental elastic displacement tensor) satisfies the équations {a) and (b) of (3.3), and the trace property

[ 5 J = £ . (3.27) For coj>0, this potential belongs to the space (H¹(f2⁺ U J ? _ ) )², therefore, from the preceeding remark, it belongs to SC ^ (see similar results for scalar wave équations in [2]). We are going hère to extend these properties to the (Hm(F))2 fonctions $ and proving that only one such

<P yields the potential u^ solution of the problem (3.3). As usual, this

<p is the unknown function of a hypersingular équation, of which we will give a variational formulation that circumvents the hypersingularity.

LEMMA 3.3 : The jump operator u G {Hl{O+ U O_ )f ^ [Ü] e (Hm(f))2 is an isomorphism between %» and {Hm{f)f for co such that a) t :> 0.

Proof : We have to prove that the problem

u G ^w such that [S] = $ (3.28) is well posed when <ol > 0. If v G $C^ then pco2 v + div <r(v) = 0 in H+ U H_, and if ü satisfies (3.28), the Green formula leads to

— a(u, v) = I (pwlu . v - <r(v). s{u))dx u a

a(v)n. $ dy . (3.29) Jr

As in the proof of theorem 3.1, one sees that the LHS of (3.29) is a continuous, coercive sesquilinear form on $ w (closed subset of (Hl(f2+ U Z2_))2, equipped with the induced norm). Thus, there is an unique iîof J^w such that (3.29) is satisfied for ail tî e 9£„. Applying again

the Green formula, one has

1

and since the normal stress application v ^ a(v)n is an isomorphism between X' a and (ƒ/" 1 / 2(f))2, it follows that [5] = $. n

It is now natural (see the recalls preceeding lemma 3.3) to call the solution of problem (3.28) the double layer elastic potential of density <p, This potential satisfies (3.29) and the coerciveness estimate (3.22) leads to

\a(u, u) \ = \ o- (u) n . <p dyf Jr

and finally the estimation (3.25) yields

( 1 + co2))33'2

H I f l j w

-

We dénote by D^M ^ the normal stress vector of u = ( y ) "¹ <p. When

<P is regular, by the expression (3.26) of ü, one gets

D „ $ = l i m o - ( - f Ta( j iy) ( x ' - y ) $ ( y ) d y < y ) ) nx, x s f . ( 3 . 3 1 )

X - » • X

That explains that D w is a hypersingular intégral operator on / \ We give in section 4 below a variational treatment of this singularity. Here, we notice that

THEOREM 3.2 : For all <o e {3m (co) = coj ^ a ^ ^ O } , D^ is an iso- morphism between (HV2(f))2 and (H~ m(f)f.

Proof : This is a simple juxtaposition of lemma 3.3 and the corollary of theorem 3.1. The application D^ can be factorised as P~loy~1, where Pw désignâtes the isomorphism between ge (H~V2(f))2 and üe 6C^^

unique solution of problem 3.3, and y the jump operator defined in corollary 3.1. • Since we will solve the intégral équation D^ <p = g by means of a variational method, the following theorem gives a more précise statement of this isomorphism.

A 2D ELASTIC CRACK PROBLEM 1 5 3 THEOREM 3.3 : The sesqui-linear form

def

K(?> h = (£>«> $> * ) for $, <A e {Hm(f)f (3.32) where the brackets stands for the duality pairing between (H~ V2(F))2 and

(HV2(F))2, satisfies the following continuity and coerciveness inequalities M ^ * > | * C — - j — M M1 / 2 i - i f|*|1 / 2 i i i i f (3.33)

. , 5

Proof : The estimate (3.33) results from (3.25) and (3.30). To obtain (3.34), we write

ba,(<pj i//) = a(y) n . [v] dyJr Jr

where u and v are respectively (y )" ¹ ^ and (y ) "^l tj/. Then, by applying the Green's formula, one gets :

(p<olu.v - <r(u). e{v))dx.

u n_

Therefore

\<o | \b„($, <P)\ - \bœ(<p, - io)<p)\^ ffleib^icp, - iojcp))

and (3.34) follows from (3.22) and (3.14). • Before ending this section, we have to corne back to the crack problem

div a{u) + pa>2u - 0 in J 3 = R2\F (a)

[cr(ü)n] =0 on r (b) (3.35) (a(u)n)+ = (<r(u)n)_ = g on F . (c)

We have signalled that (3.35a) is equivalent to (3.1a) plus the conditions

[iî] = 0 on To (3.36)

[o-(u)n] = 0 on To . (3.37) Thus, to solve (3.35), we can go through (3.3), with a condition however : to find an extension g of g on F = F U Fo such that the solution of the

problem (3.3) with data g satisfies (3.36). We will proceed differently. We notice that, if u is solution of (3.35) and

{u] on F 0 on Fo

then, if an extension g of g exists such that the corresponding problem (3.3) admits w//2+ v a_ as solution, it must be equal to £>w <p on F. Therefore, one gets

ƒ,

Jf

However, since cr{u) n^ir = g do not depend on the extension F of F and g of g, équation (3.38) should define completely <p/r when we restrict the test functions to those with support in F.

We dénote by H^sr(F) the closed subspace of functions of H* (F) with support in F, equipped with the induced norm. We suppose 3m(û>)>0.

THEOREM 3.4 : For g G (/ƒ" m(F)f, the problem (3.35) is well posed in (H^l(f2))². Its solution is the double layer potential with the density

<p solution of the variational problem

$ e (HY2(F))2 such that

fo (l ) ^{^} ^ f

where g is any extension o f g in (H~m(F))2.

Proof : Let u in (H^Y {O )f be a solution of (3.35) with g = 0. Then one can apply the Green formula to u in O⁺ U O_ and find :

(pa)2 ü . u — cr (u). ë(u)) dx = a (u) n . [u] dy .

J n+ un_ Jr

But in the RHS, a (u) n is null on F and [w] is null on Fo, thus the two sides of this equality are null. It follows that M = 0 in Q. This proves the uniqueness of the solution of (3.35).

Now, the continuity and coerciveness properties of the form b^ on {Hm{f)f remain clearly true on the closed subspace (H1/2(F))2 and the problem (3.39) is well-posed on this subspace. On the other hand, since the pairing (g, ij/} do not depend on the extension of g out of F, the solution of (3.39) do not depend either on this extension. The double layer potential of

A 2D ELASTIC CRACK PROBLEM 155 density <p, defined by (3.28), satisfies the conditions (3.36) — by the support of <p — and (3.37) — by the appartenance of u to dC'tt. Thus, it is solution of (3.35). Finally, the jump condition (3.36) implies that the norms H1 (12) of u are equal to the corresponding norms in (Hl(I2+ U O_ ))2. • Remark 3.2 : The space H1/2(F) dépends only on F. It is the same space denoted by HQQ(F) in Lions-Magenes [16]. The idea of treating the crack as a part of a closed curved has appeared in Stephan [26].

One final remark on the regularity question. It is known that the operator D^w on f is a pseudo-differential operator of order 1, and its inverse of order - 1. Consequently, if g is given in (HS(F))2 with s ^ — 1/2, and

<p the solution of the variational problem

( £ ( | , <A> V<A e (Hm(F)f (3.40) then £ E (Hs+l(f)f. Now, if g is given in (Hs(F)f with sis*- 1/2, can we conclude that the solution <p of the variational problem (3.39) is of regularity

(Hs + l(F))2l There are two difficulties for a positive answer to the question.

First, even if we choose g as an extension of g by

lr

= (0«!)/r

e (C°°(r

))

this is not sure that g G (HS(F))2 ! (we can only say that it is the case when s < 1/2).

Secondly, the variational problem (3.39) posed in a closed subspace of (Hl/2(F))2, is not extensible to the whole space as in (3.40). In f act, one can only prove the regularity property for - 1/2 =s s < 0. In a 3D situation, we refer to Stephan [27] for a proof, using pseudo-differential operator technique, and an explicit expression of the singularity of <p at the vicinity of the edges of F when g is in (H2(F))2.

3.2. The time problem

The results in this section are merely the translations of those on the preceeding section when we apply the inverse Fourier-Laplace transform to the problem (3.35).

We recall that a function f(co) is the transform of a causal temperate distribution ƒ if and only if ƒ is holomorphic in a half-plane {3m (eo ) > a}

and bounded by a polynôme in a closed half-plane included in the preceeding. This remains valid if ƒ is of values in a Hubert space. Moreover, in this case the « weak = strong » property is true for the holomorphicity, so

that in most cases it is very simple to verify this, and generally we shall omit this proof.

On the other hand, the results in section 3.1 are all formulated in the framework of L2 Sobolev spaces, with norms || . \\s and the factor I (o I T. Then, we have to précise the spatio-temp oral norms which will resuit from that in the process of the inverse transform. A convenient framework was introduced in [12]. Let cof be a positive real. We will dénote by H*l T' **'(/?" x R) the space of distributions u in Rn x R which Fourier transforms satisfy the following properties :

(i) For almost ail £ e ƒ?", w(£, co ) is analytic in {3m co > <o f}.

(ii) There exists C > 0 such that

f

It is equivalent to say that M is a causal distribution satisfying the inequality

f 2r 2 2 s |2 jRnx {R + ia>?}

= \ \CO\^2T\\Ü(., co)\\² ^ d<o < o o . ( 3 . 4 1 ) JR + iœf

The square root of the two members of (3.41) will be the norm of

o

//^ r'lü'(Rn x R), which is clearly a Hilbertian space. For a regular domain O of Rn, the space //+ Tt (°l (f2 x R ) is, as usual, the space of distributions u in 12 x R extensible to an element of //+ T' <°I(Rn x R\ with the induced norm

\\u\\^r,af.. ^ DA = i n f

xR = H

1/2

o

For F = 3/2, one defines /ƒ+ T' °>I(F x R) with the aid of an atlas as usual.

We define also H^% r' a\r x R) as the space of <p e //f'T' "/ 0(r x /?) such that its extension by 0 'm F xR belongs to H+ 'r' û'/(/; xi?). Now, we can sum up in the following theorem our results for problem (2.1).

A 2D ELASTIC CRACK PROBLEM 157 THEOREM 3.5 : Let g e (H^{+ m}' ^r' ^a \ r x R))² with co° > 0. Then

a) The problem (2.1) admits a unique solution u in (H+ T~ ' t°I (Ï2 x R))2

which is the double loyer potential of density <p e {H^{+ >T}~ ' ^<°^I (F x R))². b) If T === 1, £ fs solution of the following variational problem

(3.42)

g(x,t)^-(x,t)dy(x)dt

c) Dénote the bilinear form of (3.42) as £(<?, *A)- T'/ren (ƒ

<p e (H⁺ ' '^Ù>1 (F x /?))² on^ to.s the following weak coerciveness inequality C ( ^/ o) | | ^ | |2 7i / 2 , o , . ?( r x / ; r (3.43) Proof : If w in (ƒƒ+ T"2' W/(i7 x /?))2 is solution of (2.1) with g = 0, then its Fourier-transform ïî is solution of (3.35) for every (o G {3m co > a>f} with null data g. Thus M = 0.

Inversely, let ge (ƒƒ" 'r'W /( / " x R))2 and consider the solution ww of problem (3.35) constructed in theorem 3.4 : ww is the double layer potential with the density <p„ solution of 3.39. Since the form bw is analytic in C and coercive in { t o w > 0 } , the analyticity of gœ in {3m a> > co;0} implies that <p ^ is also analytic in this half-plane, and then the same is true for u^œt On the other hand, the estimâtes (3.34) and (3.30) give

\K\^C(a,?)\a,\\§\_^mtm (3.44)

| è^o, | ^ C ( a ,^/° ) |^û, |²| ^ | _^{i / 2 s )} (3.45) for ail <o e {3m a> > cof} (one has used the monoticity of the constants depending on coj in (3.34) and (3.30)).

Thus, part (a) is proved by the inverse Fourier transform. By using the Plancherel formula, one obtains easily part (b), from the variational problem (3.39) and part (c) from the estimâtes (3.14) and (3.22). a

158

Finally, we notice that the conditions of part (c) is satisfied as soon as g e (H~ m' 2' W/°(r x R))2 and the LHS of (3.43) is then linked to the elastic energy of the solution u of (2.1) by

(u9 t)dt where

«*•>-*ƒ, (•

(3.46)

(3.47)

In the following section, we will give a more trac table expression of the form Z?, with which the numerical experiments will be done, giving good results.

4. OBTENTION OF THE VARIATIONAL FORMULATION

This part deals with the problem of the kernel hypersingularity. By using a régularisation method developed in [6], the bilinear form can be rewritten as the sum of two parts containing weakly singular kernels. The application of this régularisation method to the obtention of the bilinear form is explained in details, in the 2D case. The 3D case is very similar and is treated in details in [4]. The bilinear form in part 3 is obtained by integrating the frequency bilinear form with respect to the frequencies

(4.1) We therefore first deal with the frequency bilinear form bw and the time domain formulation is given in the second part.

4.1. The frequency bilinear form The bilinear form is expressed as

•ƒ,

where u is solution of the problem (P%)

div er(ü) + p(o2u — 0 in /2 [o-(«)/i] = 0

[iî] = <p .

A 2D ELASTIC CRACK PROBLEM 159 We consider the derivatives in s(u) in the distributional sensé and we dénote by e(u) the function part, defined as

ê(u)= e{u)ia. (4.3)

The relation between the distribution s and its function part is given in the

LEMMA 4.1 :

e(u) = e(u)-tôr (4.4)

where tô^r is a distribution with support on the crack and t is defined as

*ij = \(<PiK} + <Pjnt). (4.5)

The proof can be found in [20]. The problem ( P | ) can now be rewritten in the distributional sensé

div a(u) + pa>2 u = 0 in O

(4.6) s(u) - s(u) = tô^r

and in f2 the Hooke's law is valid, thus we have

s{u) =Aa{u), (4.7) The problem (4.6) is transformed into a stress problem in the whole plane R2, in which only <x remains as unknown :

e(div O-(M)) + pa>2Aa = p<o2tôr . (4.8) If we dénote by G the fundamental tensor associated to (4.8), the stress tensor can be expressed as the convolution of this fundamental tensor with the right hand side :

o- = G*t8r . (4.9)

This représentation of the stress tensor is used in (4.2) to rewrite the bilinear form :

In order to reduce the singularity, we use the décomposition of G given in [6] in the gênerai anisotropic case. Ho wever, in the isotropic case, it can be directly computed and we have the

vol. 28, n° 2, 1994

THEOREM 4.1 : The fundament al tensor G can be decomposed as

G^=Dilj+pco2Ri'j (4.11)

where dénotes the spatial Fourier transform. D represents the static part, which contains the hypersingularity and R the dynamic part, which is weakly singular. Moreover, the static part D can be expressed as a fourth order differentiating operator o f a regular kernel <f> called the stress f unction. The expressions of R, D and <f> are given in (4.16)-(4.19) below.

Proof : see [6].

Notations 4.1 :

f

e^(j = - 1 if

if (ij)= (1,2) (i,y) = (2, 1) otherwise .

D

(4.12)

where Ç =

[Ds =pa>2-fJ.\£\2

is the dual variable of x in R2, and

rot a = 3ja2 - d2al = e y 3,-a,- . Expressions of R, D and $

Dp Us [- ^( A + 3 ² + /*2^

- 3 A fi | g |² 6^kl ô^u + 2 A IL (ô^kl €t èj + Sij t^k S,)]

[fJL(a«

Ski £, ij + 8U

n îp

(4.15)

(4.17) (4-18)

Remark 4.1 : 1. One can easily check, on the form of (4.18) that D represents a fourth order derivative of <f>.

A 2D ELASTIC CRACK PROBLEM 161 2. The équations (4.16) and (4,19) give the asymptotic behaviours of R and <f> when | g \ tends to infinity

and 4> = O

\i\

This shows that R and any second derivative of <j> are locally integrable.

The décomposition (4.11) is now used in (4.10) and we finally have.

THEOREM 4.2 : The frequency bilinear form has the following expression

M£. h = bi(<p, $) + bl($, h (4.20)

with

, d<p^k dé;

] FHx-y)-^(x)-^-(y)dyxdyy (4.21) Ç Ç

J pv

R

tj(x~

where the kernels F and R are weakly singular kernels expressed as some second derivatives of the kernel N :

F1 = ^ 4 - e

s

d

N (r, <* ) (4.24)

pat

- 2 A (8U a?- + «y a,2,) - /* («t f a?, + sv a^ + ski ^ + «H 3^)] N (r, a>) + , . 1 , [M («« «y + «v «„) + A «a 5y ] iV(r, « ) . (4.25)

Proof : The proof contains 3 steps 1. Use of the décomposition of G.

2. Intégration by parts of the static part, in Fourier variables.

3. Back to the space variables.

1. Use of the décomposition of G.

The expression (4.10) of the bilinear form is transformed by means of Parseval identity into

or by using (4.5) and the symmetries of G :

The décomposition (4.11) of G leads to

with

i ( £ , $) = (Dfj <p^kn^t ôⁿ <A, nj ô^r) (4.26)

The hypersingularity is now contained in the static part &i(<p, ïjf).

2. Intégration by parts of the static part.

In order to reduce the singularity, we use the remark that D coincids with a fourth-order differentiate operator of the stress function <£. This remark allows to integrate by parts and to transform this singularity into some derivatives of the unknown <p and of the test function 4f. Let us replace in (4.26) D by its expression :

(£» £ ) = (£km £jn £lq £ip %m £n £p ^q^9knl ô n & i nj ô ƒ*)

= - ((£km £ipi£miïp ^ ) r o t (<Pknör\ rot (^7«fr)) • Before concluding, we have to introducé the following identity, proved in [20]

rot (anSr) = ~ 8r (4.27)

as

valid for any scalar function a. Apply again Parseval identity, to corne back in the space domain, and use identity (4.27) to get

=~ (ekm eip Blp 0 * ^ fir j , _ i 3^ .

We thus obtain (4.21) with

F^ki = - ^{£km £}i^P Zip </> • (4.28) 3. Back to the space variables.

We now need the expressions of R and F. We thus have to compute the inverse Fourier transforms of R and of <£, which can be expressed in terms of the following quantities

Ds 1 DSDP

A 2D ELASTIC CRACK PROBLEM 163 cor

Relation (4.29) together with the expressions (4.16)-(4.19) give finally the expressions (4.24) and (4.25) of F and R. • Remark 42 : The resuit of theorem 4.1 is interesting by itself, if we want to solve the scattering problem in the frequency domain with a variational method. In most cases, régularisation technics have been developed for collocation methods. However, in the frequency domain, several similar variational formulations are available (see [3], [20]) and some numerical results can be found in [10]. There are two main advantages of the method presented hère. First, the procedure to get the bilinear form, based on the décomposition of the fundamental tensor G is very gênerai and can be extended to anisotropic media, as far as the third step of the proof is réalisable, i.e., as far as one knows how to corne back from the Fourier domain to the space domain. If this is not the case, one can always compute the inverse Fourier Transform numerically. Secondly, and this is our main purpose hère, this formulation is well suited to the time domain, thanks to the causality principle satisfied by the kernels F and /?, as we will see in the following.

4.2. The time domain bilinear form

The time domain bilinear form is obtained by integrating the frequency bilinear form (see équation 4.1). We apply again Parseval identity, written for the Fourier-Laplace transform to get its expression. It essentially consists in Computing the inverse Fourier-Laplace transform of the kernels F and R. We finally have the

THEOREM 4.3 : The space-time bilinear form has the following expression

h = b

($

h + b

(<p, h (4.30)

—— (y, t)dyxdyy dt

ds y (4.31)

164 and

E. BÉCACHE, T. HA DUONG

-²"¹'

b

($, $)= f e-

"

' f f pn

{x)n

{y)

R* Jr Jr

ÏQlïjfr-y, . ))ï'#t(x, . )) (t)^(y, t)dyxdyydt. (4.32) The inverse Fourier-Laplace of F and R are given by (4.33), (4.34) below.

(4.33) (4.34) Expressions of P^ and J5" ("/(

J ^ ( F f ) (r, t) = - 4 n2 ekn siqAnt/(r, t)

^ïJiRfj) ir, t) = P (Afjir, t) + £$ N (r, t)

where we still dénote by N (r, t) the inverse Fourier-Laplace of N (r, a> ).

N (r, t), A, P (A ) and f are defined by

I N(r, t)=NCs(r,t)-NCp(r, t)

A

C²

2 Y ²

(4.36)

A + /J. [((A + 3 ix ) (Ski 6,, + 8ti 8n) + 3 A ôkl S

- 2 A {S^kl A^tj + ôij Au)-!». {ô^kjAⁿ + ô^tj A^ik + S^ki A^Jt + 8^U A^jk)]

(4.37)

„H 1

(A + Ai) [/*(«« « y + « « « / , - ) + A a^u « y ] . (4.38)

A 2D ELASTIC CRACK PROBLEM 165 Proof : Relation (4.1) together with (4.20)-(4.22) yields

1 n JR + iui

f f 1 f / ; d(Pk d$i

jr Jr *)R + ta)i \

+ p<o²Rfj(x-y, œ) <p^k(x, <o)n^l(x)iiôij/ⁱ(y, <u ) n^}(y)) dy^xdy^y da> . (4.39) The Parseval identity, applied to the Fourier-Laplace transform is the following

J - f f(a>)g(a>)da> = [ e'2"1'f(t) g(t) dt . (4.40) Apply this identity to relation (4.39) to get the expression of the bilinear form given in (4.30)-(4.32). D As mentioned in the remark 4.1, the preceeding bilinear form is well suited to the tirne domain. In fact, the kernel F and R satisfy the causality principle, which is an essential property of the wave propagation, since it traduces the fact that the wave is travelling with a finite velocity. Of course, the fondamental solution G satisfies this causality, and this can be seen on its Fourier transform expression :

«-W ^<441)

where P (a>, £ ) is a polynomial in co and f. The fraction L = ———- is linked DP Ds

to the wave velocities and gives the causal property. From (4.41), one deduces that G is given in terms of some derivatives of the inverse Fourier transform of L, which is proportional to the kernel N⁹ and therefore possesses the causality property. Since F and R have the same form than G — i.e., they coincid with some derivatives of iV — they also possess the causality property. In [20], Nédélec proposed another décomposition of G on the following form

ifi

m*

where the first term contains the hypersmgularity and the second term is weakly singular. He also perforais an intégration by parts on the first term in order to reduce the singularity. But this time, the kernels appearing in (4.42) are given in terms of some derivatives of — whieh corresponds to the Laplace operator and not to the wave équation — and violate the causality principle. Therefore this method works in the frequency domain, but is not suited to the time domain.

5. NUMERICAL RESULTS

In the numerical application, we have considered the case of a rectilinear crack of length £, located on [0, t]. In this case, two problem have been solved : the scalar antiplane problem and the vectorial plane problem. We don't enter into details in the discrétisation since it has been done, for the antiplane crack, in [5]. Since we present simultaneously both problems (scalar and vectorial) we will « forget » the sign 7 in the following notations, unless in sections spécifie to the plane problem.

In the particular case of a rectilinear (or plane) crack, the bilinear form b^icp, if/) satisfies a coerciveness inequality without multipliying by the factor — ia>, even if coj = 0, as Ha Duong showed in [13]. We will call the resulting formulation the « first » formulation. This is the formulation directly obtained from the Boundary Intégral Equation by multiplying with a test function and by integrating on the space-time domain. The formulation presented previously (3.42), with o>^l = 0, will be called the « second » formulation.

First formulation

Cm C rco r

D<p(x,t)iff(x⁹t)dy(x)dt=\ \ g(x, t) if(x, t)dy(x)dt . Jo Jr Jo Jr

Second formulation

Jo Jr

n

*°° f èé f00

D<p(x⁹t)-£(x⁹t)dy(x)dt= , ^{w v}«, . ,

The crack [0, 2] is divided into subintervals Fj = [Xj9xj + l] of length àxj = Xj ^{+ l} — Xj⁹ with 0 = x⁰< x¹< . . . < i^j' < . . . < % = l, the time interval is divided into intervals of length àt and we define t^k = k àt. In both cases, the approximate problem is on the classical following form : Find

<ph in the finite dimensional space Vh such that

b(<ph, tf,h) = L(th) V^„eVV (5.1)

A 2D ELASTIC CRACK PROBLEM 167 For both formulations, the space V h is the space of piecewise linear functions in space and in time and the test functions are denoted by xï where i dénotes the space index and k the time index

^ , O = ? i W y * ( O ï = i iV^A, *s*o (5.2) where Nh = N -2. The unknown is decomposed on the basis of test functions :

)= £>/%

(*, O- (5.3)

It can be shown (see [4]) that the Galerkin problem (5.1) can be written as a marching-in-time scheme, which is the discrete equivalent of the convolu- tion character of the continuous intégral operator

M° ak= bk - (M1 ak~l + •*. +Mk a0) f o r k^O (5.4) where the matrices Mk and the vectors bk are defined by

xï)

Numerical Experiments

The incident wave is a plane wave propagating in the direction k with an amplitude A

u'(x, t) = Af (t-^L\ d. (5.6) The function ƒ is choosen to be linear (unless otherwise mentioned) :

ƒ (O = tH{t) (5.7) and we will call the corresponding wave a linear incident wave. In (5.6) d is a scalar equal to 1 in the antiplane problem and an unit vector corresponding to the direction of motion in the plane problem.

5-1. The Antiplane Problem

For this scalar problem, we know the asymptotic behaviour of the solution, as t tends to infinity. The solution of the antiplane problem with a linear incident wave tends to the solution of the static corresponding problem and this solution can be computed explicitly (see [25]). If 9 dénotes the

incident angle, we have :

ç> (Xj t) f- + co <p f(x) = 2 A sin é> N/Â (5.8) This case has been dealed with in [5] by using the « first » formulation. In this paper, we pointed out numerically the existence of a CFL type stability condition. We now compare this result with the « second » formulation. We use 20 boundary éléments of constant length and we will dénote by CFL the quantity — .

Ax

Figures (5.1) show the dynamical crack opening displacement (<p ) at some fixed instant compared with the static one. On figures (5.2), we have represented the évolution in time of the COD, at a fixed point on the crack (x — 0.1 ). The first axis represents t/t — i.e. the unit represents the travel time necessary for the wave to propagate from one tip to the other of the crack.

The comparison in figures (5.1 — Left) and (5.2 — Left) show that the

« second » formulation is stable for a CFL = 0.4 while the first one was unstable (the first one is stable for a CFL =s 0.3, see [5]). The first

secoad foonlatioa -

A fitst fondation - I \ static solution ••

0 0.05 0,1 0.15 0.2 U% S.3 0.35 0.4 Ö.4S 0,5 0.05 0.1 0,15 0.2 (US 0.3 0.35 0.1 0.45 O.S

Figure 5.1. — Axis : {x, <p (x)) {t being fixed). Left : Comparison between the first, the second formulation at some fixed instant and the static solution — 0 = 60° — CFL = 0.4. Right : Second formulation and static solution — 0 = 60° — CFL = 0.8 and CFL = 1.5.

A 2D ELASTIC CRACK PROBLEM

0.45

169

second formulation — first formulation •••

Figure 5.2. — Axis: (f/£, <p (0.1, t )) Left : Comparison between the fïrst and the second formulation — évolution in time — O = 60° — CFL = 0.4. Right : Time évolution of

<p computed with the second formulation — 0 = 60° — CFL - 1.5 and CFL = 3.

approximate solution oscillâtes around the static solution and the second coincids perfectly with the static solution (we have choosen enough large times). On figure (5.1 — Right), we see that the agreement of the second solution with the static solution is still very good even for large values of CFL.

Figure (5.2 — Left) show that the oscillations of the first solution appear around tlt = 2, Le. after one round trip of the wave along the crack. On the right, the time évolution of the second solution has been observed for large CFL (CFL = 1.5 and CFL = 3.) up to times equal to 15 times the travelling time along the crack. We have also carried out some other computations with different choices of CFL and ail this experiments showed that the « second » formulation is unconditionally stable. This resuit is numerically very interesting since it allows to compute the solution at large time without too much step size. Thus, this method is much less expensive. However, if we choose a too large CFL, the accuracy is not so good. This is not obvious to see it with a linear incident wave but it can be seen on the figure (5.3), where the solution corresponding to an harmonique incident wave is presented (Le.

the function ƒ is choosen harmonique).

170 E. BÉCACHE, T. HA DUONG

Figure 5.3. — Axis : (f, tp (0.2, t )) — Second formulation — Comparison between CFL = 1 and CFL = 5 for an harmonique incident wave — 0 = 60°.

5.2. The Plane Problem

In this case, there are two kinds of waves propagating 1. The Pressure waves P

\C =C^P

\d =k.

2. The Shear waves S

\C =C^S d =kx

(5.9)

(5.10) A comparison is done with the results of N. Nishimura, Q. C. Guo and S. Kobayashi (see [21]). In this paper, they solve this problem by using a collocation method. In order to compare with their resuit we multiply our

A 2D ELASTIC CRACK PROBLEM 171

2 . 5

0 . 5

1.5 -

! T 1 1 1 1 1 1 1

c

t/2a E ÜI -

1.00 2.00

- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1

Figure 5.4. — Axis : {xx/t9 cte x <p r). Incident P wave 0 = 60° — top : variational method • buttom : collocation method.